sobolev mappings, degree, homotopy classes and rational homology spheres

J Geom Anal (2012) 22:320–338DOI 10.1007/s12220-010-9194-4

Sobolev Mappings, Degree, Homotopy Classesand Rational Homology Spheres

Paweł Goldstein · Piotr Hajłasz

Received: 12 April 2010 / Published online: 11 November 2010© Mathematica Josephina, Inc. 2010

Abstract In this paper we investigate the degree and the homotopy theory of Orlicz–Sobolev mappings W 1,P (M,N) between manifolds, where the Young function P sat-isfies a divergence condition and forms a slightly larger space than W 1,n, n = dimM .In particular, we prove that if M and N are compact oriented manifolds withoutboundary and dimM = dimN = n, then the degree is well defined in W 1,P (M,N)

if and only if the universal cover of N is not a rational homology sphere, and in thecase n = 4, if and only if N is not homeomorphic to S4.

Keywords Sobolev mappings · Degree · Rational homology spheres

Mathematics Subject Classification (2000) Primary 46E35 · Secondary 46E30

1 Introduction

Let M and N be compact smooth Riemannian manifolds without boundary. Weconsider the space of Sobolev mappings between manifolds defined in the usual

Communicated by Richard Rochberg.

P.G. was partially supported by the Polish Ministry of Science grant no. N N201 397737(years 2009–2012) and P.H. was partially supported by the NSF grant DMS-0900871.

P. GoldsteinInstitute of Mathematics, Faculty of Mathematics, Informatics and Mechanics, University of Warsaw,Banacha 2, 02-097 Warsaw, Polande-mail: [email protected]

P. Hajłasz (�)Department of Mathematics, University of Pittsburgh, 301 Thackeray Hall,Pittsburgh, PA 15260, USAe-mail: [email protected]

mailto:[email protected]

mailto:[email protected]

Sobolev Mappings, Degree and Homotopy Classes 321

way: we assume that N is isometrically embedded in Euclidean space Rk and de-

fine W 1,p(M,N) to be the class of Sobolev mappings u ∈ W 1,p(M,Rk) such that

u(x) ∈ N a.e. The space W 1,p(M,N) is a subset of a Banach space W 1,p(M,Rk),

and it is equipped with a metric inherited from the norm of the Sobolev space. It turnsout that smooth mappings C∞(M,N) are not always dense in W 1,p(M,N), andwe denote by H 1,p(M,N) the closure of C∞(M,N) in the metric of W 1,p(M,N).A complete characterization of manifolds M and N for which smooth mappings aredense in W 1,p(M,N), i.e., W 1,p(M,N) = H 1,p(M,N), has recently been obtainedby Hang and Lin [22].

The class of Sobolev mappings between manifolds plays a central role in applica-tions to geometric variational problems and deep connections to algebraic topologyhave been investigated recently; see, e.g., [2–4, 7, 11, 12, 14–17, 19–22, 25, 31, 32].

In the borderline case, p = n = dimM , it was proved by Schoen and Uhlenbeck[33, 34] that the smooth mappings C∞(M,N) form a dense subset of W 1,n(M,N),i.e., W 1,n(M,N) = H 1,n(M,N), and White [38] proved that the homotopy classesare well defined in W 1,n(M,N); see also [8, 9]. Indeed, he proved that for everyu ∈ W 1,n(M,N), if two smooth mappings u1, u2 ∈ C∞(M,N) are sufficiently closeto u in the Sobolev norm, then u1 and u2 are homotopic.

If dimM = dimN = n and both manifolds are oriented, then the degree of smoothmappings is well defined. There are several equivalent definitions of the degree andhere we consider the one that involves integration of differential forms. If ω is avolume form on N , then for f ∈ C∞(M,N) we define

degf =(∫

M

f ∗ω)/(∫

N

ω

).

Since f ∗ω equals the Jacobian Jf of f (after identification of n-forms with func-tions), it easily follows from Hölder’s inequality that the degree is continuous in theSobolev norm W 1,n. Finally, the density of smooth mappings in W 1,n(M,N) allowsus to extend the degree continuously and uniquely to deg : W 1,n(M,N) → Z.

Results regarding degree and homotopy classes do not, in general, extend to thecase of W 1,p(M,N) mappings when p < n. This stems from the fact that the radialprojection mapping u0(x) = x/|x| belongs to W 1,p(Bn,Sn−1) for all 1 ≤ p < n.

A particularly interesting class of Sobolev mappings to which we can extend thedegree and the homotopy theory is the Orlicz–Sobolev space W 1,P (M,N), wherethe Young function P satisfies the so-called divergence condition

∫ ∞

1

P(t)

tn+1dx = ∞. (1.1)

In particular, P(t) = tn satisfies this condition, so the space W 1,n(M,N) is an exam-ple. However, we want the space to be slightly larger than W 1,n, and hence we alsoassume that

P(t) = o(tn) as t → ∞. (1.2)

In addition to the conditions described here we require some technical assumptionsabout P ; see Sect. 4. Roughly speaking, u ∈ W 1,P (M,N) if

∫M

P(|Du|) < ∞.

322 P. Goldstein, P. Hajłasz

A typical Orlicz–Sobolev space W 1,P discussed here contains W 1,n, and it is con-tained in all spaces W 1,p for p < n

W 1,n ⊂ W 1,P ⊂⋂

1≤p<n

W 1,p.

A fundamental example of a Young function that satisfies all required conditions,forms a strictly larger space than W 1,n, and strictly smaller than the intersection ofall W 1,p , 1 ≤ p < n, is

P(t) = tn

log(e + t).

It is easy to check that u0 �∈ W 1,P (Bn,Sn−1) if and only if the condition (1.1) issatisfied (see [17, p. 2]), so the presence of the divergence condition is necessary andsufficient for the exclusion of mappings like x/|x| that can easily cause topologicalproblems.

The class of Orlicz–Sobolev mappings under the divergence condition has beeninvestigated in connections to nonlinear elasticity, mappings of finite distortion [30],and degree theory [13, 17]. Roughly speaking, many results that are true for W 1,n

mappings have counterparts in W 1,P as well. In particular, it was proved in [17]that smooth mappings C∞(M,N) are dense in W 1,P (M,N) if P satisfies (1.1); seeSect. 4.

We say that a compact connected manifold N without boundary, dimN = n, is arational homology sphere, if it has the same de Rham cohomology groups as thesphere Sn. It has been proved in [17] that if M and N are smooth compact ori-ented n-dimensional manifolds without boundary and N is not a rational homologysphere, then the degree, originally defined on a class of smooth mappings C∞(M,N),uniquely extends to a continuous function deg : W 1,P (M,N) → Z; see also [13] forearlier results. This is not obvious, because the Jacobian of a W 1,P (M,N) mapping isnot necessarily integrable and we cannot easily use estimates of the Jacobian, like inthe case of W 1,n mappings, to prove continuity of the degree. The proof given in [17](cf. [13]) is not very geometric and it is based on estimates of integrals of differentialforms, Hodge decomposition, and the study of the so-called Cartan forms. Surpris-ingly, it has also been shown in [17] that the degree is not continuous in W 1,P (M,Sn).Thus the results of [17] provide a good understanding of the situation when N is nota rational homology sphere or when N = Sn. There is, however, a large class of ra-tional homology spheres which are not homeomorphic to Sn and it is natural to askwhat happens for such manifolds.

In this paper we give a complete answer to this problem. In our proof we do notrely on the methods of [17] and we provide a new geometric argument that avoidsmost of the machinery of differential forms developed in [17].

Theorem 1.1 Let M and N be compact oriented n-dimensional Riemannian mani-folds, n ≥ 2, without boundary, and let P be a Young function satisfying conditions(1.1), (1.2), (4.1), and (4.2). Then the degree is well defined, integer valued, and con-tinuous in the space W 1,P (M,N) if and only if the universal cover of N is not arational homology sphere.


The theorem should be understood as follows: if the universal cover of N is nota rational homotopy sphere, then deg : C∞(M,N) → Z is continuous in the normof W 1,P , and since smooth mappings are dense in W 1,P (M,N) the degree uniquelyand continuously extends to deg : W 1,P (M,N) → Z. On the other hand, if the uni-versal cover of N is a rational homology sphere, then there is a sequence of smoothmappings uk ∈ C∞(M,N) with deguk = d > 0 such that uk converges to a constantmapping (and hence of degree zero) in the norm of W 1,P .

If n = 4, Theorem 1.1 together with Proposition 2.6 give

Corollary 1.2 Let M and N be compact oriented 4-dimensional Riemannian man-ifolds without boundary and let P be a Young function satisfying conditions (1.1),(1.2), (4.1), and (4.2) with n = 4. Then the degree is well defined, integer valued, andcontinuous in the space W 1,P (M,N) if and only if N is not homeomorphic to S4.

The next result concerns the definition of homotopy classes in W 1,P (M,N).

Theorem 1.3 Let M and N be two compact Riemannian manifolds without bound-ary, n = dimM ≥ 2, and let P be a Young function satisfying the conditions (1.1),(1.2), (4.1), and (4.2). If πn(N) = 0, then the homotopy classes are well definedin W 1,P (M,N). If πn(N) �= 0, the homotopy classes cannot be well defined inW 1,P (Sn,N).

Theorem 1.3 has been announced in [17, p. 5], but no details of the proof have beenprovided. It should be understood in a similar way as Theorem 1.1. Let πn(N) = 0.Then for every u ∈ W 1,P (M,N) there is ε > 0 such that if u1, u2 ∈ C∞(M,N),‖u − ui‖1,P < ε, i = 1,2, then u1 and u2 are homotopic. Since smooth mappingsare dense in W 1,P (M,N), homotopy classes of smooth mappings can be uniquelyextended to homotopy classes of W 1,P (M,N) mappings. On the other hand, ifπn(N) �= 0, there is a sequence of smooth mappings uk ∈ C∞(Sn,N) that convergesto a constant mapping in the norm of W 1,P and such that the mappings uk are nothomotopic to a constant mapping.

The condition πn(N) = 0 is not necessary: any two continuous mappings fromCP

2 to CP1 are homotopic (in particular, homotopic to a constant mapping), even

though π4(CP1) = π4(S

2) = Z2.Theorems 1.1 and 1.3 will be proved from corresponding results for W 1,p map-

pings.

Theorem 1.4 Let M and N be two compact Riemannian manifolds without bound-ary, dimM = n, n − 1 ≤ p < n. Then the homotopy classes are well defined inH 1,p(M,N) if πn(N) = 0, and they cannot be well defined in H 1,p(Sn,N), ifπn(N) �= 0. If πn−1(N) = πn(N) = 0, then the homotopy classes are well definedin W 1,p(M,N).

The theorem should be understood in a similar way as Theorem 1.3. The lastpart of the theorem follows from the results of Bethuel [2] and Hang and Lin [22],according to which the condition πn−1(N) = 0 implies density of smooth mappingsin W 1,p(M,N), so H 1,p(M,N) = W 1,p(M,N).


Theorem 1.5 Let M and N be compact connected oriented n-dimensional Rie-mannian manifolds without boundary, n − 1 ≤ p < n, n ≥ 2. Then the degree is welldefined, integer valued, and continuous in the space H 1,p(M,N) if and only if theuniversal cover of N is not a rational homology sphere. If the universal cover of N

is not a rational homology sphere and πn−1(N) = 0, then the degree is well defined,integer valued, and continuous in W 1,p(M,N).

Again, Theorem 1.5 should be understood in a similar way as Theorem 1.1, andthe last part of the theorem follows from density of smooth mappings in W 1,p , justlike in the case of Theorem 1.4.

As a direct consequence of Theorem 1.5 and Proposition 2.6 we have

Corollary 1.6 Let M and N be compact connected oriented 4-dimensional Rie-mannian manifolds without boundary, 3 ≤ p < 4. Then the degree is well defined,integer valued, and continuous in the space H 1,p(M,N) if and only if N is not home-omorphic to S4. If N is not homeomorphic to S4 and π3(N) = 0, then the degree iswell defined, integer valued, and continuous in W 1,p(M,N).

This paper should be interesting mainly for people working in geometric analysis.Since the proofs employ quite a lot of algebraic topology, we made some effort topresent our arguments from different points of view whenever it was possible. Forexample, some proofs were presented both from the perspective of algebraic topologyand the perspective of differential forms.

The main results in the paper are Theorems 1.1, 1.3, 1.4, 1.5, Corollaries 1.2, 1.6,and also Theorem 2.1.

The paper is organized as follows. In Sect. 2 we study basic examples and prop-erties of rational homology spheres. In Sect. 3 we prove Theorems 1.4 and 1.5. Thefinal Sect. 4 provides a definition of the Orlicz–Sobolev space, its basic properties,and the proofs of Theorems 1.1 and 1.3.

2 Rational Homology Spheres

In what follows, Hk(M) will stand for de Rham cohomology groups. We say that asmooth n-dimensional manifold M without boundary is a rational homology sphereif it has the same de Rham cohomology as Sn, that is

Hk(M) ={

R for k = 0 or k = n,

0 otherwise.

Clearly M must be compact, connected, and orientable.If M and N are smooth compact connected oriented n-dimensional manifolds

without boundary, then the degree of a smooth mapping f : M → N is defined by

degf =∫M

f ∗ω∫N

ω,


where ω is any n-form on N with∫N

ω �= 0 (ω is not exact by Stokes’ theorem andhence it defines a non-trivial element in Hn(M)). It is well known that degf ∈ Z,that it does not depend on the choice of ω, and that it is a homotopy invariant.

The reason why rational homology spheres play such an important role in thedegree theory of Sobolev mappings stems from the following result.

Theorem 2.1 Let M be a smooth compact connected oriented n-dimensional man-ifold without boundary, n ≥ 2. Then there is a smooth mapping f : Sn → M ofnonzero degree if and only if the universal cover of M is a rational homology sphere.

Proof Let us notice first that if a mapping f : Sn → M of nonzero degree exists, thenM is a rational homology sphere. Indeed, clearly H 0(M) = Hn(M) = R, because M

is compact, connected, and oriented. Suppose that M has non-trivial cohomology indimension 0 < k < n, that is, there is a closed k form α that is not exact. Accordingto the Hodge Decomposition Theorem [37], we may also assume that α is coclosed,so ∗α is closed. Then, ω = α ∧ ∗α is an n-form on M such that

∫M

ω =∫

M

|α|2 > 0.

We have

degf =∫Sn f ∗ω∫

Mω

=∫Sn f ∗α ∧ f ∗(∗α)∫

Mω

= 0

which is a contradiction. The last equality follows from the fact that Hk(Sn) = 0,hence the form f ∗α is exact, f ∗α = dη. Since f ∗(∗α) is closed, we have

∫Sn

f ∗α ∧ f ∗(∗α) =∫

Sn

d(η ∧ f ∗(∗α)) = 0

by Stokes’ theorem.The contradiction proves that Hk(M) = 0 for all 0 < k < n, so M is a rational

homology sphere.On the other hand, any such mapping f factors through the universal cover M

of M , because Sn is simply connected; see [24, Proposition 1.33].

M

p

Sn

f

f

M

Clearly, M is orientable and we can choose an orientation in such a way that p isorientation preserving.

Since degf �= 0, f ∗ : Hn(M) → Hn(Sn) is an isomorphism. The factoriza-tion gives f ∗ = f ∗ ◦ p∗, and hence f ∗ : Hn(M) → Hn(Sn) = R is an isomor-phism, so M is compact. This implies that if ω is a volume form on M , then


∫Sn f ∗ω �= 0 and hence deg f �= 0. Indeed, ω defines a generator (i.e., a non-zero

element) in Hn(M) and hence f ∗ω defines a generator in Hn(Sn). If η is a volumeform on Sn, then f ∗ω = cη + dλ, c �= 0, because Hn(Sn) = R and the elements f ∗ωand η are proportional in Hn(Sn). Thus

∫Sn f ∗ω = c

∫Sn η �= 0.

The last argument can be expressed differently. Since M is compact, the numberγ of sheets of the covering is finite (it equals |πn(M)|; see [24, Proposition 1.32]). Itis easy to see that degf = γ deg f , so deg f �= 0.

Thus we proved that the mapping f : Sn → M has nonzero degree and hence M

is a rational homology sphere, by the fact obtained at the beginning of our proof.We are now left with the proof that if M is a rational homology sphere, then there

is a mapping from Sn to M of nonzero degree. Clearly, it suffices to prove that thereis a mapping f : Sn → M of nonzero degree, since the composition with the coveringmap multiplies degree by the (finite) order γ of the covering.

To this end we shall employ the Hurewicz theorem mod Serre class C of all fi-nite Abelian groups (see [26, Chap. X, Theorem 8.1], also [29], [23, Chap. 1, Theo-rem 1.8], and [35, Chap. 9, Sect. 6, Theorem 15]) that we state as a lemma.

Lemma 2.2 Let C denote the class of all finite Abelian groups. If X is a simplyconnected space and n ≥ 2 is an integer such that πm(X) ∈ C whenever 1 < m < n,then the natural Hurewicz homomorphism

hm : πm(X) → Hm(X,Z)

is a C -isomorphism (i.e., both kerhm and cokerhm lie in C ) whenever 0 < m ≤ n.

Since M is a simply connected rational homology sphere, the integral homologygroups in dimensions 2,3, . . . , n − 1 are finite Abelian groups: integral homologygroups of compact manifolds are finitely generated, thus of form Z

⊕ Zk1p1 ⊕ · · · ⊕

Zkrpr

, and if we calculate Hm(M) = Hm(M,R) with the help of the Universal Coef-ficient Theorem [24, Theorem 3.2], we have 0 = Hm(M) = Hom(Hm(M,Z),R).However, Hom(Z,R) = R

, thus there can be no free Abelian summand Z in

Hm(M,Z), and all that is possibly left is a finite Abelian group.Now we proceed by induction to show that the hypotheses of Lemma 2.2 are

satisfied. We have π1(M) = 0, thus h2 : π2(M) → H2(M,Z) is a C -isomorphism.Since H2(M,Z) is, as we have shown, in C , the group π2(M) is a finite Abeliangroup as well, and we can apply Theorem 2.2 to show that h3 is a C -isomorphismand thus that π3(M) ∈ C .

We proceed likewise by induction until we have that hn is a C -isomorphism be-tween πn(M) and Hn(M,Z) = Z. Since cokerhn = Z/hn(πn(M)) is a finite group,there exists a non-zero element k in the image of hn, and [f ] ∈ πn(M) such thathn([f ]) = k �= 0.

The generator of Hn(M,Z) = Z is the cycle class given by the whole manifold M ,i.e., hn([f ]) = k[M]; in other words, having fixed a volume form ω on M , we identifya cycle class [C] with an integer by

[C] �−→(∫

C

ω

)/(∫M

ω

),


We also recall that the Hurewicz homomorphism hn attributes to an [f ] ∈ πn(M) acycle class hn([f ]) = f∗[Sn], where [Sn] is the cycle that generates Hn(S

n,Z) = Z.Altogether,

degf =∫Sn f ∗ω∫

Mω

=∫f∗[Sn] ω∫

Mω

= k �= 0,

since f∗[Sn] = k[M]. �

To see the scope of applications of Theorem 1.1 and 1.5, it is important to under-stand how large the class of rational homology spheres is, or, more precisely, the classof manifolds whose universal cover is a rational homology sphere.

One can conclude from the proof presented above that if M is a rational homologysphere then M is a rational homology sphere as well. We will provide two differentproofs of this fact, the second one being related to the argument presented above.

Theorem 2.3 Let M be an n-dimensional compact orientable manifold withoutboundary such that its universal cover M is a rational homology n-sphere. Then

(a) M is a rational homology n-sphere as well.(b) If n is even, then M is simply connected and M = M .

Proof We start with two trivial observations: the fact that M is a rational homol-ogy sphere implies immediately that M is compact (Hn(M) = R) and connected(H 0(M) = R). This, in turn, shows that M is connected and that the number of sheetsin the covering is finite.

For any finite covering p : N → N with number of sheets γ there is not only theinduced lifting homomorphism p∗ : Hk(N) → Hk(N), but also the so-called transferor pushforward homomorphism

τ∗ : Hk(N) → Hk(N)

defined on differential forms as follows. For any x ∈ N there is a neighborhood U

such that p−1(U) = U1 ∪ · · · ∪ Uγ is a disjoint sum of open sets such that

pi = p|Ui: Ui → U

is a diffeomorphism. Then for a k-form ω on N we define

τ∗ω|U =γ∑

i=1

(p−1i )∗(ω|Ui

).

If U and W are two different neighborhoods in N , then τ∗ω|U coincides with τ∗ω|Won U ∩ W and hence τ∗ω is globally defined on N . Since

τ∗(dω)|U =γ∑

i=1

(p−1i )∗(dω|Ui

) =γ∑

i=1

d(p−1i )∗(ω|Ui

) = d(τ∗ω|U)


we see that τ∗ ◦ d = d ◦ τ∗ and hence τ∗ is a homomorphism

τ∗ : Hk(N) → Hk(N).

Clearly τ∗ ◦ p∗ = γ id on Hk(N) and hence τ∗ is a surjection.In our case N = M is a rational homology sphere and all cohomology groups

Hk(M) vanish for 0 < k < n. Since τ∗ is a surjection, Hk(M) = 0. The remainingH 0(M) and Hn(M) are equal to R by compactness, orientability, and connectednessof M .

Another proof is related to the argument used in the proof of Theorem 2.1. Bycontradiction, suppose that Hk(M) �= 0 for some 0 < k < n. Hence there is a non-trivial closed and coclosed k-form α on M . Since α is coclosed, ∗α is closed. Then

∫M

α ∧ ∗α =∫

M

|α|2 > 0,

thus

0 �= γ

∫M

α ∧ ∗α =∫

M

p∗(α ∧ ∗α) =∫

M

p∗α ∧ p∗(∗α) = 0,

because p∗α is exact on M (we have Hk(M) = 0), p∗(∗α) is closed, and hencep∗α ∧ p∗(∗α) is exact.

To prove (b), we recall that if p : N → N is a covering of order γ , then the Eulercharacteristic χN of N is equal to γχN . This can be easily seen through the trian-gulation definition of χN : to any sufficiently small simplex in N there correspond γ

distinct simplices in N , in such a way we may lift a sufficiently fine triangulation ofN to N ; clearly, the alternating sum of the number of simplices in every dimensioncalculated for N is γ times that for N .

In our case, the Euler characteristic of M and M is either 0 (for n odd) or 2 (for n

even)—this follows immediately from the fact that

χN =dimN∑i=0

(−1)i dimHi(N).

Therefore, for n = dimM even, the order of the universal covering M → M mustbe 1, and (b) follows. �

Remark 2.4 We actually proved a stronger version of part (a). If a cover (not neces-sarily universal) of a manifold M is a rational homology sphere, then M is a rationalhomology sphere, too; in the proof we never used the fact that the covering space issimply connected. However, we stated the result for the universal cover only, becausethis is exactly what we need in our applications.

Below we provide some examples of rational homology spheres.Any homology sphere is a rational homology sphere. That includes the Poincaré

homology sphere (also known as the Poincaré dodecahedral space) and Brieskorn


manifolds

�(p,q, r) = {(z1, z2, z3) ∈ C

3 : zp

1 + zq

2 + zr3 = 0, |(z1, z2, z3)| = 1

},

where 1 < p < q < r are pairwise relatively prime integers. �(2,3,5) is the Poincaréhomology sphere.

Recall that any isometry of Rn can be described as a composition of an orthogonal

linear map and a translation; the group of all such isometries, denoted by E(n), is asemi-direct product O(n)� R

n. Any discrete subgroup G of E(n) such that E(n)/G

is compact is called a crystallographic group.

Proposition 2.5 (See [36]) Let us set {ei} to be the standard basis in R2n+1 and

Bi = Diag(−1, . . . ,−1, 1︸︷︷︸i

,−1, . . . ,−1).

Let be a crystallographic group generated by isometries of R2n+1

Ti : v �−→ Bi · v + ei, i = 1,2, . . . ,2n.

Then M = R2n+1/ is a rational homology (2n + 1)-sphere.

For n = 1 this is a well known example of R3/G6; see [39, 3.5.10]. However,

π1(M) = is infinite and the universal cover of M is R2n+1, so M is an example of

a rational homology sphere whose universal cover is not a rational homology sphere.The following well known fact illustrates the difficulty of finding non-trivial ex-

amples in dimension 4:

Proposition 2.6 Let N be a compact orientable 4-manifold without boundary. Thenthe universal cover of N is a rational homology sphere if and only if N is homeomor-phic to S4.

Proof If N is homeomorphic to S4, then N = N is a rational homology sphere. Sup-pose now that the universal cover N is a rational homology sphere. Theorem 2.3tells us that N must be simply connected; therefore, H1(N,Z) = 0. Standard ap-plication of the Universal Coefficients Theorem (see, e.g., [6, Corollary 15.14.1])gives us that H 2(N,Z) has no torsion component (and thus is 0, since N is a ra-tional homology sphere) and H 1(N,Z) = 0. Poincaré duality, in turn, shows thatH 3(N,Z) ≈ H1(N,Z) = 0. The remaining H 0(N,Z) and H 4(N,Z) are Z by con-nectedness and orientability of N . Altogether, N is an integral homology sphere, andthus, by the homology Whitehead theorem [24, Corollary 4.33], a homotopy sphere.Ultimately, by M. Freedman’s celebrated result on the Generalized Poincaré Conjec-ture [10] a homotopy 4-sphere is homeomorphic to S4. Whether M is diffeomorphicto S4 remains, however, an open problem. �

One can construct lots of examples of non-simply connected rational homology4-spheres, although not every finite group might arise as a fundamental group of sucha manifold ([18, Corollary 4.4]; see also [28]).


As for higher even dimensions, A. Borel proved [5] that the only rational ho-mology 2n-sphere that is a homogeneous G-space for some compact, connected Liegroup G is a standard 2n-sphere.

Another example is provided by lens spaces [24, Example 2.43]. Given an integerm > 1 and integers 1, . . . , n relatively prime to m, the lens space, Lm(1, . . . , n) isdefined as the orbit space S2n−1/Zm of S2n−1 ⊂ C

n with the action of Zm generatedby rotations

ρ(z1, . . . , zn) = (e2πi1/mz1, . . . , e

2πin/mzn

).

Since the action of Zm on S2n−1 is free, the projection

S2n−1 → Lm(1, . . . , n)

is a covering and hence lens spaces are manifolds. One can easily prove that theintegral homology groups are

Hk(Lm(1, . . . , n)) =

⎧⎪⎨⎪⎩

Z if k = 0 or k = 2n − 1,

Zm if k is odd, 0 < k < 2n − 1,

0 otherwise.

Hence Lm(1, . . . , n) is a rational homology sphere with the universal coveringspace S2n−1.

In dimensions n > 4 there exist numerous simply connected (smooth) rationalhomology spheres; a particularly interesting set of examples are the exotic spheres,i.e., manifolds homeomorphic, but not diffeomorphic to a sphere (see, e.g., a nicesurvey article of Joachim and Wraith [27]).

3 Proofs of Theorems 1.4 and 1.5

In the proof of these two theorems we shall need the following definition and a theo-rem of B. White:

Definition 3.1 Two continuous mappings g1, g2 : M → N are -homotopic if thereexists a triangulation of M such that the two mappings restricted to the -dimensionalskeleton M are homotopic in N .

It easily follows from the cellular approximation theorem that this definition doesnot depend on the choice of a triangulation; see [22, Lemma 2.1].

Lemma 3.2 (Theorem 2, [38]) Let M and N be compact Riemannian manifolds,f ∈ W 1,p(M,N). There exists ε > 0 such that any two Lipschitz mappings g1 and g2satisfying ‖f − gi‖W 1,p < ε are [p]-homotopic.

Here [p] is the largest integer less than or equal to p. We shall also need thefollowing construction:


Example 3.3 We shall construct a particular sequence of mappings gk : Sn → Sn.Consider the spherical coordinates on Sn:

(z, θ) �→ (z sin θ, cos θ),

z ∈ Sn−1 (equatorial coordinate),

θ ∈ [0,π] (latitude angle).

These coordinates have, clearly, singularities at the north and south pole. Considerthe polar cap Ck = {(z, θ) : 0 ≤ θ ≤ 1

k} and a mapping

gk : Sn → Sn,

gk(z, θ) ={

(z, kπθ) 0 ≤ θ < 1k,

(z,π) (i.e., south pole) 1k

≤ θ ≤ π.

The mapping stretches the polar cap Ck onto the whole sphere, and maps all the restof the sphere into the south pole. It is clearly homotopic to the identity map; therefore,it is of degree 1.

The measure of the polar cap Ck is comparable to k−n, |Ck| ≈ k−n. It is also easyto see that the derivative of gk is bounded by Ck.

Let 1 ≤ p < n. Since the mappings gk are bounded and gk converges a.e. to theconstant mapping into the south pole as k → ∞, we conclude that gk converges to aconstant map in Lp . On the other hand, the Lp norm of the derivative Dgk is boundedby

∫Sn

|Dgk|p ≤ Ckp|Ck| ≤ C′kp−n → 0 as k → ∞,

and hence gk converges to a constant map in W 1,p .

Proof of Theorem 1.4 Assume that πn(N) = 0 and f ∈ H 1,p(M,N), with n − 1 ≤p < n. By Lemma 3.2 any two smooth mappings g1 and g2 sufficiently close to f are(n − 1)-homotopic, that is, there exists a triangulation T of M such that g1 and g2,restricted to the (n − 1)-dimensional skeleton Mn−1 of T, are homotopic.

Let H : Mn−1 × [0,1] → N be a homotopy between g1 and g2 and let � be anarbitrary n-simplex of T. We have defined a mapping H from the boundary of � ×[0,1] to N : it is given by g1 on � × {0}, by g2 on � × {1} and by H on ∂� × [0,1].However, ∂(�×[0,1]) is homeomorphic to Sn. Since πn(N) = 0, any such mappingis null-homotopic and extends to the whole � × [0,1]. In such a way, simplex bysimplex, we can extend the homotopy between g1 and g2 onto the whole M .

We showed that any two smooth mappings sufficiently close to f in the norm ofW 1,p are homotopic, and we may define the homotopy class of f as the homotopyclass of a sufficiently good approximation of f by a smooth function. Hence homo-topy classes can be well defined in H 1,p(M,N).

If in addition πn−1(N) = 0, then according to [22, Corollary 1.7], smooth map-pings are dense in W 1,p(M,N), so W 1,p(M,N) = H 1,p(M,N) and thus homotopyclasses are well defined in W 1,p(M,N).


In order to complete the proof of the theorem we still need to show that ifπn(N) �= 0, we cannot define homotopy classes in W 1,p(Sn,M), n − 1 ≤ p < n, ina continuous way. As advertised in the Introduction, we shall construct a sequenceof smooth mappings that converge in W 1,p(Sn,N) to a constant mapping, but arehomotopically non-trivial.

If πn(N) �= 0, we have a smooth mapping G : Sn → N that is not homotopic toa constant one. Hence the mappings Gk = G ◦ gk are not homotopic to a constantmapping, where gk was constructed in Example 3.3. On the other hand, since themappings gk converge to a constant map in W 1,p , the sequence Gk also converges toa constant map, because composition with G is continuous in the Sobolev norm. �

The condition πn(N) = 0 is not necessary for the mappings g1 and g2 to be ho-motopic, as can be seen from the following well known proposition.

Proposition 3.4 Any two continuous mappings CP2 → CP

1 are homotopic, whileπ4(CP

1) = Z2.

Sketch of a proof of Proposition 3.4 Since CP1 = S2, we have π4(CP

1) = π4(S2) =

Z2 (see [24, p. 339]).The space CP

2 can be envisioned as CP1 = S2 with a 4-disk D attached along

its boundary by the Hopf mapping H : S3 → S2. Therefore the 2-skeleton (CP2)(2)

consists of the sphere S2.Suppose now that we have a mapping φ : CP

2 → CP1. If φ is not null-homotopic

(i.e., homotopic to a constant mapping) on (CP2)(2), then its composition with the

Hopf mapping is not null-homotopic either, since H generates π3(S2) = Z. Then φ

restricted to the boundary S3 of the 4-disk D is not null-homotopic and cannot beextended onto D, and thus onto the whole CP

2. Therefore, we know that φ restrictedto the 2-skeleton of CP

2 is null-homotopic.This shows that φ is homotopic to a composition

CP2 p−→ CP

2/(CP2)(2) = S4 → CP

1 = S2.

It is well known that π4(S2) = Z2 and that the only non-null-homotopic mapping

S4 → S2 is obtained by a composition of the Hopf mappingH : S3 → S2 and of its suspension �H : S4 → S3 (see [24], Corollary 4J.4 andfurther remarks on EHP sequence). Then, if φ is to be non-null-homotopic, it mustbe homotopic to a composition

CP2 p−→ S4 �H−−→ S3 H−→ S2.

The first three elements of this sequence are, however, a part of the cofibration se-quence

S2 ↪→ CP2 p−→ S4 �H−−→ S3 → �CP

2 → ·· · ,

and as such, they are homotopy equivalent to the sequence

S2 ↪→ CP2 → CP

2 ∪ C(S2) ↪→ CP2 ∪ C(S2) ∪ C(CP

2) ≈ CP2 ∪ C(S2)/CP

2,


since attaching a cone to a subset is homotopy equivalent to contracting this sub-set to a point (by C(A) we denote a cone of base A). One can clearly see that inthe above sequence the composition of the second and third mapping and the ho-motopy equivalence at the end, CP

2 → CP2 ∪ C(S2)/CP

2, is a constant map; thusthe original map �H ◦ p : CP

2 → S3 is homotopic to a constant map, and so isH ◦ �H ◦ p : CP

2 → S2, the only candidate for a non-null-homotopic mapping be-tween these spaces. �

Proof of Theorem 1.5 This proof consist of two parts:In the first one, suppose N is such that its universal cover N is a rational homology

sphere; n = dimM = dimN . We shall construct an explicit example of a sequence ofmappings in H 1,p(M,N), n − 1 ≤ p < n of a fixed, non-zero degree, that convergein H 1,p-norm to a constant mapping.

The manifold M can be smoothly mapped onto an n-dimensional sphere (take asmall open ball in M and map its complement into the south pole). This mappingis clearly of degree 1; we shall denote it by F . Notice that F , by construction, is adiffeomorphism between an open set in M and Sn \{south pole}. Next, let us considera smooth mapping G : Sn → N of non-zero degree, the existence of which we haveasserted in Theorem 2.1.

We define a sequence of mappings Fk : M → N as a composition of F , mappingsgk given by Example 3.3 and G:

Fk = G ◦ gk ◦ F. (3.1)

The degree of Fk is equal to degG, thus non-zero and constant. On the other hand,we can, exactly as in the proof of Theorem 1.4, prove that Fk converges in W 1,p toa constant map. Since Fk converges a.e. to a constant map, it converges in Lp . Asconcerns the derivative, observe that |DFk| ≤ Ck and DFk equals zero outside theset F−1(Ck) whose measure is comparable to k−n. Hence

∫M

|DFk|p ≤ Ckp |F−1(Ck)| ≤ C′kp−n → 0 as k → ∞.

and thus the convergence to the constant map is in W 1,p .This shows that the mappings Fk can be arbitrarily close to the constant map, so

the degree cannot be defined as a continuous function on H 1,p(M,N), n−1 ≤ p < n.Now the second part: we shall prove that, as long as the universal cover N of N is

not a rational homology sphere, the degree is well defined in H 1,p(M,N), n − 1 ≤p < n. To this end, we will show that it is continuous on C∞(M,N) equipped withthe W 1,p distance and therefore it extends continuously onto the whole H 1,p(M,N).In fact, since the degree mapping takes value in Z, we need to show that any twosmooth mappings that are sufficiently close in H 1,p(M,N) have the same degree.

Consider two smooth mappings g,h : M → N that are sufficiently close to a givenmapping in H 1,p(M,N). By Lemma 3.2 of White, g and h are (n − 1)-homotopic,i.e., there exists a triangulation T of M such that g and h are homotopic on the(n − 1)-skeleton Mn−1. Let H : Mn−1 × [0,1] → N be the homotopy, H(x,0) =g(x), H(x,1) = h(x) for any x ∈ Mn−1.


Let us now look more precisely at the situation over a fixed n-simplex � of thetriangulation T. We have just specified a continuous mapping H on the boundary of� × [0,1]: it is given by g on � × {0}, by h on � × {1}, and by H on ∂� × [0,1].

Note that ∂(�×[0,1]) is homeomorphic to Sn; therefore, the mapping H : ∂(�×[0,1]) → N is of degree zero, since N is not a rational homology sphere. Thus wemay fix the orientation of ∂� × [0,1], so that

∫�

Jg =∫

�

Jh +∫

∂�×[0,1]JH . (3.2)

Recall that the Jacobian Jf of a function f : M → N , with fixed volume forms μ

on M and ν on N , is given by the relation f ∗ν = Jf μ, and degf = (∫M

Jf dμ)/

(∫N

dν) = |N |−1∫M

Jf dμ. Therefore, by summing up the relations (3.2) over all then-simplices of T we obtain

|N |degg =∫

M

Jg dμ

=∑

�n∈T

(∫�n

Jh dμ +∫

∂�n×[0,1]JH dμ

)

=∫

M

Jh dμ +∑

�n∈T

∫∂�n×[0,1]

JH dμ

= |N |degh +∑

�n∈T

∫∂�n×[0,1]

JH dμ.

We observe that every face of ∂�n ×[0,1] appears in the above calculation twice, andwith opposite orientation, thus

∑�n∈T

∫∂�n×[0,1] JH cancels to zero, and degg =

degh. Finally, if the universal cover of N is not a rational homology sphere andπn−1(N) = 0, then H 1,p(M,N) = W 1,p(M,N) by [22, Corollary 1.7] and hencethe degree is well defined in W 1,p(M,N). �

4 Orlicz–Sobolev Spaces and Proofs of Theorems 1.1 and 1.3

We shall begin by recalling some basic definitions of Orlicz and Orlicz–Sobolevspaces; for a more detailed treatment, see, e.g., [1, Chap. 8] and [17, Chap. 4].

Suppose P : [0,∞) → [0,∞) is convex, strictly increasing, with P(0) = 0. Weshall call a function satisfying these conditions a Young function. Since we want todeal with Orlicz spaces that are very close to Ln, we will also assume that P satisfiesthe so-called doubling or �2-condition:

there exists K > 0 such that P(2t) ≤ K P(t) for all t ≥ 0. (4.1)

This condition is very natural in our situation and it simplifies the theory a great deal.Under the doubling condition the Orlicz space LP (X) on a measure space (X,μ) is


defined as a class of all measurable functions such that∫X

P (|f |) dμ < ∞. It is aBanach space with respect to the so-called Luxemburg norm

‖f ‖P = inf

{k > 0 :

∫X

P (|f |/k) ≤ 1

}.

We say that a sequence (fk) of functions in LP (X) converges to f in mean, if

limk→∞

∫X

P (|fk − f |) = 0.

It is an easy exercise to show that under the doubling condition the convergence inLP is equivalent to the convergence in mean.

For an open set � ⊂ Rn we define the Orlicz–Sobolev space W 1,P (�) as the space

of all the weakly differentiable functions on � for which the norm

‖f ‖1,P = ‖f ‖P +m∑

i=1

‖Dif ‖P

is finite. For example, if P(t) = tp , then W 1,P = W 1,p . In the general case, con-vexity of P implies that P has at least linear growth and hence LP (�) ⊂ L1

loc(�),

W 1,P (�) ⊂ W1,1loc (�). Using coordinate maps one can then easily extend the defini-

tion of the Orlicz–Sobolev space to compact Riemannian manifolds, with the result-ing space denoted by W 1,P (M).

Now let M and N be compact Riemannian manifolds, n = dimM . We shall beinterested in Orlicz–Sobolev spaces that are small enough to exclude, for M = Bn,N = Sn−1, the radial projection x �→ x/|x|. As shown in [17, p. 2], to exclude suchprojection it is necessary and sufficient for P to grow fast enough to satisfy the so-called divergence condition, already announced (see (1.1)) in the Introduction:∫ ∞

1

P(t)

tn+1= ∞.

The function P(t) = tn satisfies this condition, but we are not interested in Orlicz–Sobolev spaces that are too small (that are contained in W 1,n(M,N)), so we imposean additional growth condition on P , already stated (see (1.2)) in the Introduction:

P(t) = o(tn) as t → ∞.

This condition is important. In the case of P(t) = tn the degree and homotopy resultsfor mappings between manifolds are well known, and in this instance the resultshold without any topological assumptions about the target manifold N . Our aim isto extend the results beyond the class W 1,n, and the dependence on the topologicalstructure of N is revealed only when the Orlicz–Sobolev space is larger than W 1,n,so we really need this condition.

In order to have C∞(M,N) functions dense in LP (M,N), we need yet anothertechnical assumption: that the function P does not ‘slow down’ too much; more pre-cisely, that

the function t−αP (t) is non-decreasing for some α > n − 1. (4.2)


This condition is also natural for us. We are interested in the Orlicz–Sobolev spacesthat are just slightly larger than W 1,n, so we are mainly interested in the situationwhen the growth of P is close to that of tn, and the above condition requires less thanthat. Note that it implies

W 1,P (M,N) ↪→ W 1,α(M,N) ↪→ W 1,n−1(M,N).

The condition (4.2) plays an important role in the proof of density of smoothmappings.

Lemma 4.1 [17, Theorem 5.2] If the Young function P(t) satisfies the diver-gence condition (1.1), doubling condition (4.1), and growth condition (4.2), thenC∞(M,N) mappings are dense in W 1,P (M,N).

In particular, in order to extend continuously the notions of degree and homotopyclasses to W 1,P (M,N), it is enough to prove that they are continuous on C∞(M,N)

endowed with W 1,P norm (provided P satisfies all the hypotheses of Lemma 4.1).Note that density of smooth mappings in W 1,P (M,N) and the embedding

W 1,P (M,N) ↪→ W 1,n−1(M,N) implies that W 1,P (M,N) ↪→ H 1,n−1(M,N).

Proof of Theorem 1.3 Let M , N , and P be as in the statement of Theorem 1.3. As-sume also that πn(N) = 0. By the above remark, it is enough to prove that if twosmooth mappings f,g : M → N are sufficiently close to a W 1,P (M,N) mapping in‖ · ‖1,P -norm then they are homotopic. However, by the inclusion W 1,P (N,M) ↪→H 1,n−1(N,M), we know that f and g are close in H 1,n−1(N,M), and by Theo-rem 1.4 they are homotopic.

If πn(N) �= 0, we can construct, like in the proof of Theorem 1.4, a sequence ofnon-null-homotopic mappings convergent to a constant mapping in W 1,P (Sn,N).Indeed, the mappings Gk constructed as in the proof of Theorem 1.4 converge a.e.to a constant mapping, so they converge to a constant mapping in mean and hencein LP . On the other hand, the derivative of Gk = G ◦ gk is bounded by Ck and hence

∫Sn

P (|DGk|) ≤ P(Ck)|Ck| ≤ C′P(k)k−n → 0 as k → ∞

by (1.2) and the doubling condition. Hence DGk converges to zero in mean and thusin LP . �

Proof of Theorem 1.1 If the universal cover of N is not a rational homology sphere,the continuity of the degree mapping with respect to the Orlicz–Sobolev norm, as inthe previous proof, is an immediate consequence of Theorem 1.5: our assumptionson P give us an embedding of W 1,P (M,N) into H 1,n−1(M,N).

What is left to prove is that if the universal cover of N is a rational homologysphere, the degree cannot be defined continuously in W 1,P (M,N). If Fk = G◦gk ◦F

are the mappings defined as in the proof of Theorem 1.5, then Fk converges a.e. to aconstant map, so it converges in LP . As concerns the derivative, |DFk| is bounded by


Ck and different from zero on a set F−1(Ck) whose measure is comparable to k−n,so ∫

M

P(|DFk|) ≤ P(Ck)|F−1(Ck)| ≤ C′P(k)k−n → 0 as k → ∞

by (1.2) and the doubling condition. Hence DFk converges to zero in mean and thusin LP . �

Acknowledgements The authors would like to thank all who participated in a discussion posted onAlgebraic Topology Discussion List (http://www.lehigh.edu/~dmd1/post02.html) in the thread Questionabout rational homology spheres posted by Hajłasz in 2002. The authors also thank J. Kedra for pointingout Proposition 3.4. And last, but not least, the authors acknowledge the kind hospitality of CRM at theUniversitat Autònoma de Barcelona, where the research was partially carried out.

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sobolev mappings, degree, homotopy classes and rational homology spheres

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